Q: If a long hot streak is less likely than a short hot streak, then doesn’t that mean that the chance of success drops the more successes there are?

One of the original questions was: I understand “gambler’s fallacy” where it is mistaken to assume that if something happens more frequently during a period then it will be less frequently in the future. Example: If I flip a coin 9 times and each time I get HEADS, than to assume that it is more “probable” that the 10th flip will be tails is a incorrect assumption.

I also understand that before I begin flipping that coin in the first place, the odds of getting 10 consecutive HEADS is a very big number and not a mere 50/50.

My question is: Is it more likely?, more probable?, more expectant?, or is there a higher chance of a coin turning up TAILS after 9 HEADS?

Physicist: Questions of this ilk come up a lot. Probability and combinatorics, as a field study, are just mistake factories. In large part because single words massively change the difference between two calculations, not just in the result but in how you get there. In this case the problem word is “given”.

Probabilities can change completely when the context, the “conditionals”, change. For example, the probability that someone is eating a sandwich is normally pretty low, but the probability that a person is eating a sandwich given that there’s half a sandwich in front of them is pretty high.

To understand the coin example, it helps to re-phrase in terms of conditional probabilities. The probability of flipping ten heads in a row, , is . Not too likely.

The probability of flipping tails given that the 9 previous flips were heads is a conditional probability: P(T | 9H) = P(T) = 1/2.

In the first situation, we’re trying to figure out the probability that a coin will fall a particular way 10 times. In the second situation, we’re trying to figure out the probability that a coin will fall a particular way only once. Random things like coins and dice are “memoryless”, which means that previous results have no appreciable impact on future results. Mathematically, when A and B are unrelated events, we say P(A|B) = P(A). For example, “the probability that it’s Tuesday given that today is rainy, is equal to the probability that it’s Tuesday” because weather and days of the week are independent. Similarly, each coin flip is independent, so P(T | 9H) = P(T).

The probability of the “given” may be large or small, but that isn’t important for determining what happens next. So, after the 9th coin in a row comes up heads everyone will be waiting with bated breath (9 in a row is unusual after all) for number ten, and will be disappointed exactly half the time (number 10 isn’t affected by the previous 9).

This turns out to not be the case when it comes to human-controlled events. Nobody is “good at playing craps” or “good at roulette”, but from time to time someone can be good at sport. But even in sports, where human beings are controlling things, we find that there still aren’t genuine hot or cold streaks (sans injuries). That’s not to say that a person can’t tally several goalings in a row, but that these are no more or less common than you’d expect if you modeled the rate of scoring as random.

For example, say Tony Hawk has already gotten three home runs by dribbling a puck into the end zone thrice. The probability that he’ll get another point isn’t substantially different from the probability that he’d get that first point. Checkmate.

Notice the ass-covering use of “not substantially different”. When you’re gathering statistics on the weight of rocks or the speed of light you can be inhumanly accurate, but when you’re gathering statistics on people you can be at best humanly accurate. There’s enough noise in sports (even bowling) that the best we can say with certainty is that hot and cold streaks are not statistically significant enough to be easily detectable, which they really need to be if you plan to bet on them.

12 Responses to Q: If a long hot streak is less likely than a short hot streak, then doesn’t that mean that the chance of success drops the more successes there are?

That’s not actually a good example of independence, as it has been demonstrated that weather and days of the week aren’t independent; it’s more likely to rain later in the week due to build up of ozone and carbon monoxide in the local atmosphere from commuters, with the likelihood falling back to normal after the weekend as the levels fall due to reduced driving. It’s far from insignificant too; in the Eastern Seaboard it’s 22% more likely to rain on a Saturday than on a Monday.

Great explanation, but I don’t think you really answered the question! The original asker accepts that each coin flip has no effect on subsequent flips, but is struggling with the following apparent paradox: you’ve flipped a coin and seen it come up 9 times heads, which is pretty unlikely. Now for the 10th flip: 10 heads in a row is an even MORE unlikely event than 9! So I BELIEVE that the previous flips haven’t altered the coin and it still has a fair chance of landing either way. But if the next flip is either going to keep us at 9 heads or take us to 10 heads, then if 9 is more likely than 10 doesn’t that mean we MUST be more likely to see a tails??

I think what causes us to look at it this way, intuitively, is our innate tendency to separate

past events from future events. Since the first 9 flips already happened, we feel that all the unlikeliness of the 10-head streak is now wrapped in that one last flip. Really, thigh, ANY of the previous flips COULD have been tails – the unlikeliness of the whole 10 heads is equally distributed across the 10 flips. Think of it this way: is 5 heads, then a tail, then 4 more heads more or less likely than 9 heads and a tail at the end? Any specific combination of heads and tails is equally as likely as any other. So 9 heads and a tail is equally as likely as 9 heads and a head – in any specific order. It’s a very subtle move to instead look at a group of flips and talk about the low probability that NONE of them will come up tails. The paradox comes in when, in the heat of the moment, you incorrectly apply corect reasoning about a GROUP of flips to the single flip at the very end, just because it hasn’t happened yet.

Does anybody else agree that the FEELING of the gambler’s fallacy still kind of needs to be addressed beyond just reasserting that the events are indeed independent? (Even if you think I’m wrong about where that feeling comes from?)

On a vaguely related note to Doc G’s post, I’d also say that while in this presentation you stated a priori that it was a fair coin, from a Bayesian perspective if all you know about a coin is that it has come up heads 9 times in a row, that’s pretty good evidence that it’s not a fair coin, which shifts your priors such that in that sense P(H|9H)>0.5. 😛

@Doc G
The probability of 10 heads is .5*.5*.5*…*.5=(1/2)^10.
The probability of 9 head is (1/2)^9. On the last flip, there is a 50/50 chance of either a heads or a tails. Thus, the probability for 10 head is (1/2)^10. The probability of 9 heads and 1 tails is also (1/2)^10 (assuming the tails comes at the end or the beginning, I’m not counting all the possibilities for a single tails to show up randomly somewhere, in other words I’m only counting the two permutations, not all the different combinations).

There might exist a hidden condition working to make this hot/cold streak.
In sports, air pressure, humidity might alter a strike.
There is no way to consider every condition already known and there are some other hidden acting.
Supose we ignore low air pressure makes the ball go further, and game have been taken half in low and half of them in high.
Some particular technique might succed more in low than in high, or viceversa.
There must be a mathematical way to detect a hidden condition.